1869 would be post-civil war and early reconstruction Era. So let's think this out. I'm hardly an expert on late 19th century American or Latin politics, but here goes.
Most territories were, in the eyes of settlers and of the American government, unoccupied except for disorganized Indians. However, the Dominicans were a relatively modern society of European descent, with cities, infrastructure and relatively contemporary agriculture.
So, based on that, its likely that the Dominican Republic would have ascended to statehood relatively quickly, like Texas. The nativism probably doesn't get out of hand until the 1880's and 1890's, when the South got enough vileness under its belt to pass Jim Crow laws, and the rest of the country was willing to go around. On the other hand 1869 were the aftermath of the civil war, the dying days of slavery, and a new era which included the 14th Amendment.
This gives them two senators, and perhaps a handful of Reps. I don't know how the house of representatives worked back then. I would assume that there are roughly 60 Senators, so on the surface, two might not make that big a difference. Of course, this might not be true if the Senate or the House was a strongly contested battlefield where even a few seats might swing things one way or the other. I'm not clear, however, that such a battlefield existed.
On the other hand, the plantation/cotton economy of the South had been disrupted, no one quite knew in what direction it might go. So there might have been considerable interest in and enthusiasm for the Dominican as an economic model for the development of southern agriculture or crops. The Dominican economy might have gone flush with land purchases, land speculation, and innovative new crops. So contemplate a boom and bust cycle, and the decline of small landholdings in favour of larger land acquisitions. Also, more turnover, at least for a while, in agricultural land ownership. There might be a period where the Dominicans were on the verge of being an engine of the American economy, before the Southern economies reasserted themselves.
The Dominicans become the center of US caribbean trade, likely increased trade volume with Puerto Rico, Cuba, Jamaica and the Antilles. But likely more interactions with and more underlying conflicts with the colonial powers, Spain, Netherlands, Britain and France. Earlier or increased involvement with Latin America.
For United States policy, the largest effect would probably be a large state right in the middle of the Caribbean. That would have to have an effect on military priorities. I would say a major investment in naval facilities in the Dominican Republic, and a larger US naval investment overall. More ships, larger ships, and a caribbean bias. As in the modern era, it may well be that Dominican naval facilities become a political and economic plum to be defended and supported at all costs. Possibly some Dominican naval shipbuilding, and consequent industrialisation? That might come later. Assuming that the navy is the critical military arm, perhaps the navy comes to have an increasing proportion of Dominicans or latins?
Not a lot of anglo settlement. That's all going west, mostly. And the Dominicans are already pretty full. On the other hand, English is the language of the navy, and of business and commerce. I suspect that both state and federal governments will make efforts to anglicize the place. English becomes the favoured language of elites and middle classes, is taught in schools, etc.
As a result of a larger US naval and economic presence in the Caribbean, and as a result of local political agitation by the Dominicans for something like 'lebensraum' (the same sort of local agitation had lead to the Texan sitatuation, the Mexican American war, and then the Indian Wars) larger I'm thinking that the Spanish American War comes earlier. It's likely driven by Dominicans who feel that their economic interests lie in expanding American power in the Caribbean.
How much earlier? In our timeline, the Spanish American war was 1899. The Dominicans join the US in 1869. That's only thirty years. Let's assume minimum 10 to 20 years to accumulate enough cloud and nagging to get things going. So between 1879 and 1889. Around the time of manifest destiny and the settlement of the west. On the other hand, its also around the time of the pushback against reconstruction.
Puerto Rico and Cuba are taken. In our timeline, keeping or letting Cuba go was a near thing, and resulted in a compromise - conquer but don't keep. This time, its kept. Probably neither become a state immediately, and statehood is more controversial. But by this time, the Dominicans are in congress and pushing hard. So statehood comes within a decade. By 1900 at latest, Cuba and Puerto Rico are states.
American investment runs in part through the Dominican Republic, which is somewhat anglicized. Trade with Puerto Rico and Cuba, and Dominican activities there (attempting to make these states economic hinterlands) results in a revival of latin language and culture in the Dominican.
Will Haiti be absorbed? That's the 64,000 dollar question, and the short answer is likely no. It's 99% black. The American congress will balk, and increasingly so as time goes on. For the same reason, the last big island, Jamaica, will probably be safe. On the other hand, I'm not sure that we wouldn't see the economic colonization of Haiti - the establishment of major foreign owned landholdings, plantation economy, sharecropping, and ready use of US military power to hold the place both under control and arms length.
One question about the earlier Spanish American war. Will the Phillipines and Guam be taken? Will they be kept? The Phillipinos fought a pretty fierce insurrection in our timeline. Several possibilities emerge: 1) The Phillipnes remain the last Spanish bastion. 2) The Phillipines become Independent. 3) The Phillipines become an American possession moving to Statehood. 4) The Phillipines fall under the sway of some other Imperial power. 5) What happens in our timeline - colonial possession.
A Dominican influenced foreign policy may focus much more on the Caribbean and Central America, may be less interested in Asia.
Speaking of which, does the Panama Canal happen the same way? Would the Canal go through Nicaragua, earlier? I can see the Dominicans pushing the project big time, seeing all sorts of spin offs for them for it. Does the new Panama get to become a US state? By this time, there will already be three latin states.
Does WWI happen the same way? I believe that it was a near thing for the US to go in with Germany or Britain. Do Dominican ambitions in the Caribbean tip America towards Germany? Britain and France sue for peace, and there is the humiliating loss of the British or or French and British possessions, from Jamaica to the Bahamas, Bermuda, the Antilles and Guyana?
At this point, the Caribbean becomes a truly American lake, with American possessions throughout.
But the situation in Europe is massively butterflied.
Just some thoughts.
